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Full text of "Treatise On Analysis Vol-Ii"

264      XIV   INTEGRATION IN LOCALLY COMPACT GROUPS

(cf. Section 14.2, Problem 4(d)). In particular, we have

a-iCVrCA)) => Vr(ow!(A))
for all r > 0, in the notation of Section (3.6). (Observe that Vr(A) = A + B'(0; r).)

8.    Let H c R" be a hyperplane passing through the origin 0, and let r be a rotation
transforming H into R""1. For each universally measurable bounded subset A of RM,
put cr«(A) = T~l • crn_i( T- A) (the " Steiner symmetrization " of A with respect to H).

(a)    Suppose that A is closed and contained in a closed ball B with center 0. Show that,
if W is an open subset of the sphere S which is the boundary of B and if W n A = 0,
then aH(A) does not intersect W nor the image of W under the symmetry with respect

toH.

(b)    Deduce from (a) that, under the same hypotheses, there exist finitely many
hyperplanes Hi,...,Hr passing through 0, such that the set aHraHr.i •••aHl(A)
is contained in B and does not meet S unless A = B (use the compactness of S).

(c)    Suppose that A is compact. If B0 is the closed ball with center 0 such that
Afl(A) = An(B0), show that there exists a sequence (Hm) of hyperplanes passing through
0, such that the sequence of compact sets Am = aHttl <rHm_i' *' crwi(A) tends to B0 in
the topology defined in Section 3.16, Problem 3. (Use the result of this problem, by
showing first that the sequence (Am) can be assumed to have a limit A' such that
A' <= B'(0; R), where R is the greatest lower bound of the radii of closed balls with
center 0 containing a transform of A under the composition of a finite number of
Steiner symmetrizations aH with respect to hyperplanes H passing through 0. Then
argue by contradiction and use (b) above to show that B'(0; R) = B0.)

9.    (a)   Let A be a nonempty compact subset of the space R" endowed with the Euclidean
scalar product. For each unit vector u e R", put h(A', u) = sup(* | u), and b(A; u) =

ueA

/z(A; w) + /z(A; —u) (the width of A in the direction u). The least upper bound of the
numbers b(A't u) as u varies on the unit sphere Srn_i is the diameter 8(A) of A (Section
6.3, problem 2). For any pair of compact sets A, B and any real number a > 0 we
have h(A + B; u) = //(A; u) + /z(B; w) and h(<x.A; u) — aA(A; u), from which it
follows that 8(A + B) ^ 8(A) + 8(B).

(b)    Let s be a finite sequence ((a/, C//))i $j*m of pairs in which the otj are real numbers
^0 such that]T aj = 1, and the Uj are rotations about 0 (i.e., elements of SO(/z, R)).

The rotational mean of A corresponding to the finite sequence 5 is defined to be the
compact set p*(A) =£ o,£/,(A). We have An(p5(A)) £ Art(A) (Section 14.1, Problem

4(d)). For each u e Sn_i, show that

h(ps(A); (/)•=£ *jh(A; Uj-\u)).

(c)    Let A be a nonempty compact set contained in a closed ball B with center 0,
and let S be the frontier of B (a sphere). Show that if there exists a nonempty open
subset W of S which does not meet A, then there exists a rotational mean ps(A) such
that ps(A) c B and S o ps(A) = 0 (same method as Problem 7; use the compactness
ofS).

(d)    Let R be the greatest lower bound of radii of closed balls with center 0 containing
a transform of A under the composition of a finite number of rotational means. If
Bo = B'(0;R), show that there exists a sequence of sets Ap==pJppSp_l •••psl(A)
tending to B0, relative to the distance defined in Section 3.16, Problem 3. (Argue byssociated with w, y, w, respectively (Section 13.18,